Sequence: an ordered list of a finite or infinite quantity of numbers.
Usually, sequences are built according to some kind of system such as 2, 4, 8, 16, 32, ... The three dots at the end indicate that the sequence will be continued infinitely according to the same system. As the order of the numbers in a given sequence is predetermined, the numbers can be enumerated. Thus, the set of numbers in an infinite sequence is always ►countable.
One of the most famous infinite sequences of numbers is the one made up of the Fibonacci numbers. This sequence starts out with 0 and 1; each subsequent Fibonacci number equals the sum of its two immediate predecessors:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765...
You will encounter the Fibonacci numbers in many mathematical and technical areas, including the Golden Ratio (Binet's formula), Pascal's triangle (sums of the diagonals), in combinatorics, and even the structure of leaves in nature.
The sum of a sequence is what we call a series. Imagine you added the number 1 infinitely many times:
1 + 1 + 1 + 1 + 1 + 1 + ...
Obviously, this sum is infinite. And if we replace the number 1 by any other finite number - even a number as small as 0.00000001 - then we will have the same outcome. An infinite series of constant numbers is always infinite.
Now, what happens if the numbers do not remain constant but change? Let us assume that the numbers steadily increase, as they do in the Fibonacci sequence:
0 + 1 + 1 + 2 + 3 + 5 + 8 + ...
This sum can be once again dissolved into an infinite sum containing only digits 1 and 0:
0 + 1 + 1 + (1+1) + (1+1+1) + (1+1+1+1+1) + (1+1+1+1+1+1+1+1) + ...
Obviously, the sum of an increasing infinite sequence is always infinite as well. This suggests that the sum of a decreasing infinite sequence should also be infinite. Astonishingly, however, this is not always the case.
The most simple series of decreasing numbers is the so-called geometric series:
In this progression, each number simply equals half of the preceding number. The second most famous series of decreasing numbers is the harmonic series:
Both series are sums of infinitely many numbers and look very much alike. Nonetheless, there is a significant difference between them: the former is a finite sum, the latter is not. The mathematician ►John Wallis was the first to systematically explore infinite sequences and series (in his book Arithmetica Infinitorum), and it was he who first published this astonishing result.
Let us consecutively add the individual numbers of the geometric series and look at what number is approximated by the total sum:
Each consecutive interim sum in this series comes closer to the number 2. Though we add infinitely many numbers, the result does not at all equal an infinite number. Rather, the total sum of this infinite geometric series equals the finite number 2!
We have a different case in the interim sums of the harmonic series. These climb slowly but steadily beyond the number 2 and eventually beyond any finite number. Mathematicians therefore conclude that the geometric series has a limit, namely the number 2, while the harmonic series does not. The fact that infinitely many additions sometimes result in a finite value already puzzled the ancient Greeks and inspired ►Zeno's paradoxes.
Another interesting infinite series is the following:
1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 ...
This is Grandi's series, named after the mathematician who published it in 1703; it corresponds to the puzzle of Thompson's lamp. Imagine you had a very durable desk lamp with a toggle switch the flicking of which consecutively turns the lamp on and off. Now is the lamp turned on or off after flicking the switch infinitely many times? In other words: What is the limit of the above series? There are two intuitively plausible answers:
1) The limit is 0, and the lamp will be off. By using parentheses I can rewrite the above series thus: (1 - 1) + (1 - 1 )+ (1 - 1) + (1 - 1) ... = 0 + 0 + 0 + 0 ... = 0. For if I flick the switch infinitely many times then this is the same as if I had flicked it twice an infinite number of times. But flicking the switch twice does not affect the on- or off-status of the lamp. It remains off.
2) False. Rather, the limit is 1, and the lamp will be on. For I can rewrite the above series thus: 1 + (- 1 + 1) + (- 1 + 1) + (- 1 + 1) . = 1 + 0 = 1. The lamp is turned on with the first flicking; the subsequent double-flicking does not alter its status.
Since both arguments are equally strong, though their conclusions obviously contradict each other, the status of the lamp is regarded as indeterminate. Though the sum of the sequence of flickings never gets beyond the number 1, it does not have a limit. The philosopher Gottfried Leibniz even offered a third alternative solution: according to him, the limit is ½, since the infinite number of additions between 0 and 1 created some kind of new mean value. Contemporary mathematicians, however, do not share this view. Undoubtedly, Leibniz would have offered an unconventional solution to the lamp problem as well if electric light had existed during his time.